• Volume 116, Issue 4

November 2006,   pages  373-551

• Preface

• Schrödinger Operators on the Half Line: Resolvent Expansions and the Fermi Golden Rule at Thresholds

We consider Schrödinger operators $H= -d^2/dr^2+V$ on $L^2([0,∞))$ with the Dirichlet boundary condition. The potential 𝑉 may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum of 𝐻 is classified, and asymptotic expansions of the resolvent around zero are obtained, with explicit expressions for the leading coefficients. These results are applied to the perturbation of an eigenvalue embedded at zero, and the corresponding modified form of the Fermi golden rule.

• Corners of Normal Matrices

We study various conditions on matrices 𝐵 and 𝐶 under which they can be the off-diagonal blocks of a partitioned normal matrix.

• Nice Surjections on Spaces of Operators

A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni [9] of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections on $\mathcal{K}(X, Y)$ for Banach spaces 𝑋, 𝑌. We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result for $\mathcal{L}(X, Y)$ thereby proving an analogue of the result from [9] for $L^p(1 < p ≠ 2 < ∞)$ spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from [8].

• Invariants for Normal Completely Positive Maps on the Hyperfinite $II_1$ Factor

We investigate certain classes of normal completely positive (CP) maps on the hyperfinite $II_1$ factor $\mathcal{A}$. Using the representation theory of a suitable irrational rotation algebra, we propose some computable invariants for such CP maps.

• Remarks on $B(H) \otimes B(H)$

We review the existing proofs that the min and max norms are different on $B(H)\otimes B(H)$ and give a shortcut avoiding the consideration of non-separable families of operator spaces.

• Generalized Unitaries and the Picard Group

After discussing some basic facts about generalized module maps, we use the representation theory of the algebra $\mathscr{B}^a(E)$ of adjointable operators on a Hilbert $\mathcal{B}$-module 𝐸 to show that the quotient of the group of generalized unitaries on 𝐸 and its normal subgroup of unitaries on 𝐸 is a subgroup of the group of automorphisms of the range ideal $\mathcal{B}_E$ of 𝐸 in $\mathcal{B}$. We determine the kernel of the canonical mapping into the Picard group of $\mathcal{B}_E$ in terms of the group of quasi inner automorphisms of $\mathcal{B}_E$. As a by-product we identify the group of bistrict automorphisms of the algebra of adjointable operators on 𝐸 modulo inner automorphisms as a subgroup of the (opposite of the) Picard group.

• The Planar Algebra of a Semisimple and Cosemisimple Hopf Algebra

To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.

• Gromov-Witten Invariants and Quantum Cohomology

This article is an elaboration of a talk given at an international conference on Operator Theory, Quantum Probability, and Noncommutative Geometry held during December 20--23, 2004, at the Indian Statistical Institute, Kolkata. The lecture was meant for a general audience, and also prospective research students, the idea of the quantum cohomology based on the Gromov-Witten invariants. Of course there are many important aspects that are not discussed here.

• Representations of Homogeneous Quantum Lévy Fields

We study homogeneous quantum Lévy processes and fields with independent additive increments over a noncommutative -monoid. These are described by infinitely divisible generating state functionals, invariant with respect to an endomorphic injective action of a symmetry semigroup. A strongly covariant GNS representation for the conditionally positive logarithmic functionals of these states is constructed in the complex Minkowski space in terms of canonical quadruples and isometric representations on the underlying pre-Hilbert field space. This is of much use in constructing quantum stochastic representations of homogeneous quantum Lévy fields on Itô monoids, which is a natural algebraic way of defining dimension free, covariant quantum stochastic integration over a space-time indexing set.

• Stochastic Integral Representations of Quantum Martingales on Multiple Fock Space

In this paper a quantum stochastic integral representation theorem is obtained for unbounded regular martingales with respect to multidimensional quantum noise. This simultaneously extends results of Parthasarathy and Sinha to unbounded martingales and those of the author to multidimensions.

• Malliavin Calculus of Bismut Type without Probability

We translate in semigroup theory Bismut’s way of the Malliavin calculus.

• Construction of some Quantum Stochastic Operator Cocycles by the Semigroup Method

A new method for the construction of Fock-adapted quantum stochastic operator cocycles is outlined, and its use is illustrated by application to a number of examples arising in physics and probability. The construction uses the Trotter–Kato theorem and a recent characterisation of such cocycles in terms of an associated family of contraction semigroups.

• On Equivariant Dirac Operators for $SU_q(2)$

We explain the notion of minimality for an equivariant spectral triple and show that the triple for the quantum 𝑆𝑈(2) group constructed by Chakraborty and Pal in [2] is minimal. We also give a decomposition of the spectral triple constructed by Dabrowski et al [8] in terms of the minimal triple constructed in [2].

• Subject Index

• Author Index

• # Proceedings – Mathematical Sciences

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